Previous research has shown sketching to be useful to students solving math problems. The present study examines which aspects of middle school students’ sketching are related to, or predict, successfully answering math problems. The effects of individual differences in cognitive factors – working memory, spatial ability, and prior math knowledge – on answer accuracy are also analyzed. Stepwise regression analysis indicates that prior math knowledge and the inclusion of numerical representations of key problem relationships in sketches positively predict answer accuracy, whereas including irrelevant relationships in a sketch is associated with lower answer accuracy. Methodological implications for future research are discussed.

Indigo is widely used in textile industries for denim garments dyeing and is mainly produced by chemical synthesis which, however, raises environmental sustainability issues. Bio-indigo may be produced by fermentation of metabolically engineering bacteria, but current methods are economically incompetent due to low titer and the need for an inducer. To address these problems, we first characterized several synthetic promoters in E. coli and demonstrated the feasibility of inducer-free indigo production from tryptophan using the inducer-free promoter. We next coupled the tryptophan-to-indigo and glucose-to-tryptophan pathways to generate a de novo glucose-to-indigo pathway. By rational design and combinatorial screening, we identified the optimal promoter-gene combinations, which underscored the importance of promoter choice and expression levels of pathway genes. We thus created a new E. coli strain that exploited an indole pathway to enhance the indigo titer to 123 mg/L. We further assessed a panel of heterologous tryptophan synthase homologs and identified a plant indole lyase (TaIGL), which along with modified pathway design, improved the indigo titer to 235 mg/L while reducing the tryptophan byproduct accumulation. The optimal E. coli strain expressed 8 genes essential for rewiring carbon flux from glucose to indole and then to indigo: mFMO, ppsA, tktA, trpD, trpC, TaIGL and feedback-resistant aroG and trpE. Fed-batch fermentation in a 3-L bioreactor with glucose feeding further increased the indigo titer (≈965 mg/L) and total quantity (≈2183 mg) at 72 h. This new synthetic glucose-to-indigo pathway enables high-titer indigo production without the need of inducer and holds promise for bio-indigo production.